Compton Shift and Scattering Calculator
Calculate wavelength shift, scattered wavelength, incident and scattered photon energy, and recoil electron kinetic energy from either incident wavelength or photon energy plus scattering angle.
Assumption:
electron initially at rest and effectively free.
Results
Δλ shift
—
λC(1 − cos θ)
Scattered wavelength λ′
—
λ + Δλ
Incident photon energy
—
E = hc/λ
Scattered photon energy
—
E′ = hc/λ′
Recoil electron kinetic energy
—
Energy transferred
Uses λC = h/(mec) = 2.42631023867 pm for an electron.
Scattering diagram
Worked solution
- Enter a starting wavelength or photon energy and angle to generate a step-by-step substitution.
Compton shift by angle
| Angle | 1 − cos θ | Δλ | λ′ | E′ | Energy loss |
|---|---|---|---|---|---|
| Run a calculation to compare common scattering angles. | |||||
Formula reference
Compton scattering describes how a photon loses energy when it scatters from a free electron. This calculator solves the common homework and lab quantities: wavelength shift, scattered wavelength, incident and scattered photon energy, and recoil electron kinetic energy.
Shift equation
Δλ = λC(1 − cos θ)
θ is the photon scattering angle from 0° to 180°.
Scattered wavelength
λ′ = λ + Δλ
The outgoing photon wavelength is longer when θ > 0°.
Photon energy
E = hc/λ
Using hc = 1.239841984 keV·nm.
Electron constants
λC = h/(mec) = 2.42631023867 pm
Maximum electron shift = 2λC = 4.85262047734 pm.
Recoil energy
Ke = Ein − Eout
This is the photon energy transferred to the electron in the simple model.
Variables
λ: incident wavelength; λ′: scattered wavelength; Δλ: shift; θ: scattering angle; me: electron rest mass.
Compton highlights
λC is tiny
The Compton wavelength h/(mec) ≈ 2.43 pm. Even big-angle scattering barely nudges visible light but meaningfully shifts X-rays.
Scale
θ drives the loss
Δλ peaks at 180°. Forward scatter (θ≈0°) barely changes energy.
Geometry
Energy inverse to λ
E = hc/λ. A positive Δλ means lower photon energy and momentum—transferred to the electron.
Energy transfer
Quantum signature
Compton’s experiment showed light carries momentum like a particle, not just a wave—key evidence for photons.
History
Backscatter creates a floor
At θ = 180°, the scattered photon has the lowest possible energy for a given input λ. This “backscatter edge” is used in detector calibration.
Calibration
Methodology and verification
Constants used
λC = 2.42631023867 pm and hc = 1.239841984 keV·nm.
Calculation assumptions
Electron is initially at rest and effectively free; binding, Doppler broadening, and cross-section effects are not modeled.
Reviewed
Last reviewed: June 8, 2026.
Privacy
Inputs and results are calculated in your browser and are not sent to a server by this tool.
FAQ
What is the maximum Compton shift?
For an electron initially at rest, the maximum shift occurs at θ = 180° and equals 2λC, or 4.85262047734 pm.
Why does the initial wavelength not affect Δλ?
The shift equation Δλ = λC(1 − cos θ) depends on angle and electron mass. The starting wavelength affects λ′ and the photon energies, but not the added wavelength shift.
Why does visible light barely shift?
Visible wavelengths are hundreds of nanometers, while λC is only about 0.002426 nm. Even the maximum shift is a tiny fraction of visible-light wavelength.
How do I calculate scattered photon energy?
Calculate Δλ, add it to the incident wavelength to get λ′, then use E′ = hc/λ′. If you start from energy, convert to wavelength first with λ = hc/E.
When does the free-electron approximation work?
It works best when the photon energy is large compared with electron binding energies and the electron can be treated as stationary before the collision.
Does the formula work for protons or nuclei?
The same form can be written with the target particle mass, but this calculator uses the electron mass. A proton or nucleus has a much smaller Compton wavelength, so the shift is much smaller.
How is Compton shift different from Compton wavelength?
The Compton wavelength λC is the constant h/(mec). The Compton shift Δλ is the actual angle-dependent increase λC(1 − cos θ).